Aufgaben:Exercise 1.1Z: Non-redundant Binary Source

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Dirac-shaped source signal

Any digital source can be completely described by its source symbol sequence

$$\langle q_\nu \rangle = \langle \hspace{0.05cm}q_0 \hspace{0.05cm}, q_1 \hspace{0.05cm}, q_2 \hspace{0.05cm}, ... \hspace{0.05cm} \rangle$$

Contrary to the theory part, here the control variable  $\nu$  starts with zero. If each individual symbol  $q_\nu$  originates from the symbol set  $\{\rm L, \ H\}$, it is called a binary source.

Using the symbol spacing  $T$,  one can also characterize the source symbol sequence  $\langle q_\nu \rangle$  in an equivalent way by the Dirac-shaped source signal

$$q(t) = \sum_{(\nu)} a_\nu \cdot {\rm \delta} ( t - \nu \cdot T)$$

which rather corresponds to a system-theoretic approach. Here, we denote  $a_\nu$  as the amplitude coefficients.

• In the case of a binary unipolar digital signal transmission, holds:

$$a_\nu = \left\{ \begin{array}{c} 1 \\ 0 \\ \end{array} \right.\quad \begin{array}{*{1}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} q_\nu = \mathbf{H} \hspace{0.05cm}, \\ q_\nu = \mathbf{L} \hspace{0.05cm}. \\ \end{array}$$

• Correspondingly, in the case of a bipolar system:

$$a_\nu = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right.\quad \begin{array}{*{1}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} q_\nu = \mathbf{H} \hspace{0.05cm}, \\ q_\nu = \mathbf{L} \hspace{0.05cm}. \\ \end{array}$$

The diagram shows the Dirac-shaped source signal  $q(t)$  of a binary source. It is known from this source that it is redundancy-free. This statement is quite relevant for solving the problems.

Notes:

• The exercise belongs to the chapter  System Components of a Baseband Transmission System.
• Reference is made in particular to the section  Descriptive variables of the digital source.

• In the literature, the two possible binary symbols are usually designated as  $\rm L$  and  $\rm 0$. 
• To avoid the somewhat confusing mapping  $a_\nu = 1$  for  $q_\nu =\rm 0$  and  $a_\nu = 0$  for  $q_\nu =\rm L$,  we use the symbols  $\rm L$  ("Low") and  $\rm H$  ("High") in our learning tutorial.

Questions

1

What is the symbol distance  $T$?

$T \ = \ $ $\ \rm µ s$

2

What is the bit rate  $R$ output by the source?

$R \ = \ $ $\ \rm kbit/s$

3

Is this representation unipolar or bipolar?

	The symbol sequence is unipolar.
	The symbol sequence is bipolar.

4

What is the source symbol  $q_2$?

	$q_2 = \rm L$,
	$q_2 = \rm H$.

5

What is the symbol probability  $p_{\rm H} = {\rm Pr}(q_\nu = \rm H$)?

$p_{\rm H} \ = \ $

Solution

Solution

(1)  According to the diagram, the distance between two symbols is $\underline{T = 2\ \rm µ s}$.

(2)  With this redundancy-free binary source – and only with such a source – the bit rate is $R = 1/T\hspace{0.15cm}\underline{=500 \ \rm kbit/s}$.

(3)  The possible amplitude coefficients are $\pm 1$. Therefore, the given symbol sequence is bipolar.

(4)  The amplitude coefficient $a_2$ can be read at $2T = 4 \ \rm µ s$.

• With bipolar mapping, it follows that $a_2 = -1$ for symbol $q_2 \hspace{0.15cm}\underline {=\rm L}$.

(5)  Even if the diagram suggests otherwise for the short time interval shown here:

• For a redundancy-free binary source, in addition to the statistical independence of the symbols, $p_{\rm H} = p_{\rm L}\hspace{0.15cm}\underline{ = 0.5}$ (equally probable symbols) must also hold.